3.1. Baseline S&OP Model: Order Placement, Flexibility, and Inventory Control

The baseline model considers highly unpredictable customer demand with no significant trend or seasonality. In such cases, customer demand arrival is modeled as a uniformly distributed random parameter. High forecasting errors, which are common with long procurement times that can lead to unreliable demand predictions, are also incorporated in the model ^2,6. Predicted demand is calculated as the sum of the actual demand arrival and a uniformly distributed forecasting error, as shown in Eq. \(\eqref{eq:1}\).

As illustrated in Fig. 1, the rolling planning horizon comprises a frozen horizon (\(t\) to \(t+F-1\)), a flexible horizon (\(t+F\) to \(t+F+4\)), and a free horizon beginning at \(t+F+5\). The actual demand arrival (\(D_t^a\)) at period \(t\) is allocated to future periods \(j\) within the flexible horizon in fractional quantities (\(O_{tj}\)), as defined in Eqs. \(\eqref{eq:2}\) and \(\eqref{eq:3}\). These fractions are aggregated at each replanning cycle and become actual orders \(d_t^a\) (yellow bars in Fig. 1). Order-placing factors (\(x_{j-t}\)) represent the demand arrival rates estimated based on industrial data ⁶. These customer orders are placed independently of the predicted demand (red) and parts availability. The order placement factors used in this study are listed in Table 2.

At the time fence (\(t+F\)), the maximum number of orders that the sales department can accept subject to manufacturing capacity limits are defined by the sales constraint (\(d_t^{\textit{max}}\)), as illustrated in Fig. 1. This constraint, which includes an allowance above the predicted demand, is calculated using the flexibility degree in Eq. \(\eqref{eq:4}\) ⁶. It defines the flexibility range (light green) beyond which customer orders are delayed or lost. Delayed orders are backorders caused by capacity limits, i.e., the sales constraint at a future time fence period, and should not be confused with execution-level delivery delays in the current period, which are mitigated through backup supply. As the planning horizon rolls forward, the remaining actual orders to be produced, i.e., the production order quantity (\(N_t=\min\{d _t^a;d_t^{\textit{max}}\}\)), enter the frozen horizon, where no further changes are allowed. Under this strict time-based rule, no new orders are placed into frozen periods, even if the production quantities are below the available capacity. Any remaining predicted demand (\(d_t^{\textit{pr}}=\max\{0;d_t^p-d_t^a\}\)) is removed from the frozen horizon.

The flexibility degree determines the sales constraint, whereas the safety stock ratio functions as a supply buffer that accounts for customer order variability within the flexible horizon. In practice, these variables are determined through negotiation among the sales, manufacturing, and logistics departments and remain fixed throughout each planning cycle. In contrast, the linear policy defined in Eqs. \(\eqref{eq:5}\) and \(\eqref{eq:6}\) enables dynamic adjustment of both variables at each planning cycle and has been demonstrated to outperform the static policy ⁷. The decision variables are the absolute and relative coefficients of flexibility degree \((\textit{FR}^{\alpha },\textit{FR}^{\beta})\) and the safety stock ratio \((\textit{SS}^{\alpha},\textit{SS}^{\beta})\). The parameter \(\varPhi_t\) in Eq. \(\eqref{eq:7}\) represents the ratio of the expected demand to the historical moving average of the predicted demand over \(M\) periods. When expected demand exceeds the moving average (\(\varPhi_t>1\)), demand is overestimated, resulting in a lower flexibility degree and a lower safety stock ratio to avoid unused capacity and excess inventory. Conversely, when \(\varPhi_t<1\), demand is underestimated, leading to higher flexibility and safety stock to prevent delayed orders and stockouts.

3.2. Effects of Frozen Horizon Length Adjustments

The frozen horizon length determines the time fence position and fixes orders within those periods, thereby stabilizing manufacturing and logistics operations. Fig. 2 illustrates a sample plan with actual orders (\(d_t^{ak})\) for planning cycle \(k\) and period \(t\). Blue-shaded periods denote the frozen horizon, while the remaining periods represent the flexible horizon. The time fence periods are marked in bold within boxes. Together, these sections constitute the total planning horizon (PH), which rolls forward by one period at each planning cycle. Orange-shaded periods are supply arrival times, which are discussed in the next subsection.

In the automotive industry, especially where imported vehicles often outcompete locally manufactured models, customer impatience is higher. Longer frozen horizons lead to extended delivery times, thereby increasing the risk of lost sales, whereas shorter horizons enhance responsiveness and increase tolerance for backorders. Delayed orders and lost sales are determined by the sales constraint and lost sales threshold (\(Z\)) (Fig. 1). The threshold represents customer impatience beyond which customers cancel their orders ⁸. In this study, the lost sales threshold is modeled relative to the frozen horizon length to capture the effects of customer impatience in S&OP. This relationship is estimated based on the customer impatience level toward backordering in the case industry and is used in the simulation model (Table 3). Actual orders exceeding the sales constraint, but within the threshold are delayed to the next period, as in Eq. \(\eqref{eq:8}\), whereas any excess orders beyond the threshold are lost, as in Eq. \(\eqref{eq:9}\).

The effects of frozen horizon adjustments on delivery time flexibility and plan stability necessitate the introduction of both objectives within the S&OP model. The formulation for the average delivery time is presented below, and incorporation of plan stability objectives is discussed in the subsequent subsection.

The model measures the average promised delivery time assigned to outstanding orders within the current planning horizon, considering frozen and flexible horizon lengths, delayed orders, and lost sales. As shown in Eq. \(\eqref{eq:10}\), it is calculated as the weighted average of the actual orders from the upcoming period (\(t+1\)) to the end of the planning horizon (\(t+F+4\)), where each order is weighted by its respective period \(m\), divided by the total number of actual orders. This represents the average due date of all orders in the current planning horizon waiting to be fulfilled averaged over the simulation length \(H\).

3.3. Incorporating Plan Stability Objectives

To evaluate schedule changes within the rolling-horizon planning framework, explicit KPIs for plan instability and local and global nervousness are incorporated in the S&OP model. Schedule changes emerge from frozen horizon length adjustments, sales constraints imposed on actual demand at the time fence, and the aggregation of customer orders within the flexible horizon. Using the sample plan shown in Fig. 2 (Section 3.2), these KPIs are defined and introduced in the model as follows.

Capturing intra-cycle actual order variability between periods is crucial because it quantifies how fluctuations in production quantities affect capacity utilization, production flow, and supply order consistency within the rolling planning horizon. Plan instability therefore measures the average difference (smoothness) in scheduled quantities between consecutive periods within the same planning cycle ^12,22. As shown in Fig. 2 and Eq. \(\eqref{eq:11}\), for Planning Cycle 5 at Period 4, it is computed as the sum of absolute differences in actual orders between two consecutive periods from Period 4 to Period 11 divided by the total number of periods minus one.

A plan may be internally smooth within a cycle yet undergo frequent adjustments across cycles, increasing the coordination burden across functions, reducing productivity and worker morale, and causing supplier dissatisfaction. Plan nervousness measures changes in order timing or quantity in the current or lower-level schedules due to minor adjustments at higher or similar planning levels ^12,22. Local nervousness for Planning Cycle 5 at Period 4 is computed as the sum of the absolute changes that the actual order quantity (\(d_4^{a5}\)) has undergone from Planning Cycles 1 to 5 in Period 4, divided by its current value (Eq. \(\eqref{eq:12}\). Global nervousness extends this calculation across the entire planning horizon by summing the absolute changes the actual orders have undergone from Planning Cycles 1 to 5 for each period from \(t=4\) to \(t=11\), divided by the total of actual orders in the current cycle (Eq. \(\eqref{eq:13}\). These KPIs are averaged over the simulation length.

3.4. Modeling Procurement Time Uncertainty

Procurement time in global supply chains is typically long and highly uncertain. Lead times for different orders can become independent of each other, making the modeling difficult because the arrival times can cross despite the sequence in which they were placed. Most studies in the literature did not allow order crossing, assuming that the interval between orders is usually large enough ³⁶. In this study, we consider a short normal supply interval (NSI) and allow the possibility of order crossing due to higher uncertainty in distant sourcing. Moreover, multi-sourcing introduces additional variations in supplier lead times. Thus, orders placed in a certain period may arrive either earlier or later than the expected arrival time (\(t+L\)), leading to concurrent or missed arrivals and resulting in excess inventory or shortages.

The supplier lead times (\(L_{it}\)) are modeled as uniform distributions (Fig. 2). The figure shows the supply orders from two suppliers placed at \(k=1,3,5,\dots\) (\(\textit{NSI}=2\)). It shows overlapping (orange-shaded) and non-overlapping (light orange) supply arrivals as well as order crossings between cycles. The expected arrival lead time \(L\) is determined by averaging the minimum and maximum lead times of each supplier (\(L_i^{\textit{min}}\) and \(L_i^{\textit{max}}\)) within their respective distributions, as shown in Eq. \(\eqref{eq:14}\). This value is used to determine the anticipated inventory and normal supply required in the next subsection.

3.5. Integration of Capacity-Constrained Supply Order Allocation

The case company procures vehicle parts as completely knocked down (CKD) units based on the demand plan. In the baseline S&OP model, the normal supply at \(t+L\) (Eq. \(\eqref{eq:16}\) is calculated from the expected demand, the anticipated inventory at \(t+L-1\) (Eq. \(\eqref{eq:15}\), and the safety stock ratio (determined in Section 3.1). The safety stock ratio provides a buffer for the expected changes in actual orders from period \(t+F+1\) to \(t+L\) ⁶.

However, the model assumes that the exact normal supply quantity arrives within a deterministic lead time and without capacity constraints. In practice, parts may be sourced from multiple suppliers with uncertain lead times and limited capacities. This study extends the model by computing the anticipated inventory and normal supply for the expected arrival period \(t+L\) (determined using Eq. \(\eqref{eq:14}\) and splitting the normal supply required among suppliers using the newly introduced order allocation percentage decision variables (\(q_i\)), subject to capacity constraints and lead-time uncertainty.

The capacity of each supplier is modeled with minimum and maximum values based on its standard order quantity for initiating purchases and maximum production capacity. Accordingly, the allocated orders for each supplier subject to their respective capacity limits are determined as shown in Eq. \(\eqref{eq:17}\). Because each supplier has an uncertain lead time, deviations can occur between the required normal supply at \(t+L\) and the realized supply due to different arrival times (\(t+L_{it}\)). This can result in missed or concurrent arrivals, causing shortages or excess inventory that reflect the degree of lead-time overlap within the same planning cycle (Fig. 2). Similarly, lead-time overlap between planning cycles, i.e., order crossing, can occur depending on the normal supply interval.

The realized normal supply at period \(t\) is compiled as in Eq. \(\eqref{eq:18}\) considering the concurrently arriving orders placed at \(t-L_{it}\) from different suppliers and planning cycles. The parameter \(c\) is used to determine the range of possible previous planning cycles (\(k\)) at which separate orders may cross and arrive concurrently. It is calculated as \(c=\max\{L_i^{\textit{max}}\}-\min \{L_i^{\textit{min}}\}\). If \(\textit{NSI}>c\), order crossing does not occur. The simulation model compiles the total realized normal supply (\(n_t^R\)) from each supplier’s normal supply arrival (\(n_{it}\)) ordered over the range of \(k\) cycles as defined in the equation.

Backup supplies are modeled following previous studies. When a shortage occurs in period \(t\), a primary backup supply is ordered and arrives within the same period, as in Eq. \(\eqref{eq:19}\) ⁶. In addition, a negotiated backup, as in Eq. \(\eqref{eq:20}\), is used to improve customer service and total profit. This supply is scheduled for period \(t+F\) after negotiation with impatient customers who might otherwise cancel their orders, and is estimated from sales at risk of being lost and the probability of customer agreement ⁹. Both backup supply types are not subject to supply capacity constraints and incur higher unit costs than normal supply. The resulting net actual inventory at the end of period \(t\) is determined using Eq. \(\eqref{eq:21}\).

3.6. Optimization of the Integrated S&OP System

The model also incorporates conventional KPIs, such as total profit and the percentages of delayed orders and lost sales. They are introduced prior to formulating the integrated S&OP optimization model. The price factor in Eq. \(\eqref{eq:22}\) is included in the profit calculation to compensate for the negotiated backup supply costs covered by customers who accepted the negotiated delivery, thereby avoiding backorders. Manufacturing cost considers overtime when the production order quantity (\(N_t\)) exceeds the normal manufacturing capacity (\(N\)), as in Eq. \(\eqref{eq:23}\). Based on the previously determined variables and unit cost data, the average total profit over the simulation length is calculated using Eq. \(\eqref{eq:24}\) ⁹. Customer service performance is measured through delayed-order and lost-sales percentages, defined as the ratio of delayed orders and lost sales to the total actual orders across the simulation length, as shown in Eqs. \(\eqref{eq:25}\) and \(\eqref{eq:26}\) ⁶.

The integrated S&OP model considers total profit, delayed-order and lost-sales percentages, average delivery time, plan instability, and nervousness as inherently conflicting objectives that require multi-objective optimization. Decision variables include the absolute and relative coefficients of flexibility degree and safety stock ratio, and order allocation percentages. Increasing flexibility improves service levels but may lead to excessive backup supply costs and lower profits. Higher safety stocks reduce shortages but increase inventory holding risks under demand and lead-time uncertainty. Longer frozen horizons enhance schedule stability but may extend delivery times, raise costs, and increase lost sales, reflecting the inherent trade-offs among the objectives.

To manage these trade-offs, the \(\varepsilon\)-constraint method is applied, scalarizing the optimization problem into a single objective with one primary objective and the others as bounding constraints. This approach ensures global optimality of the primary objective while maintaining Pareto optimality of the remaining objectives whose values typically lie near their respective \(\varepsilon\)-constraint bounds. This method is used because it effectively handles multiple conflicting objectives, allows decision makers to specify or validate acceptable bounds for secondary objectives, and enables efficient exploration of Pareto-efficient solutions.

Maximization of total profit is treated as the primary objective, while the \(\varepsilon\)-constraint objectives include maximum delayed-order and lost-sales percentages, delivery time, plan instability, and local and global nervousness denoted as DOP\(^{\textit{max}}\), LSP\(^{\textit{max}}\), \(T^{\textit{max}}\), \(I^{\textit{max}}\), LN\(^{\textit{max}}\), and GN\(^{\textit{max}}\), respectively. These values can be tuned to generate multiple viable solutions; however, careful consideration is required when dealing with a large number of objectives, as in this study ³⁷. Accordingly, the \(\varepsilon\)-vector values were determined through iterative simulation experiments and subsequently validated by company experts. This process ensured feasible solutions and balanced trade-offs across conflicting objectives in the absence of predetermined company thresholds. The selected values are presented in Section 4.3.

The optimization model is formulated using Eqs. \(\eqref{eq:27}\)–\(\eqref{eq:35}\). Eq. \(\eqref{eq:27}\) defines the primary objective function, while Eqs. \(\eqref{eq:28}\)–\(\eqref{eq:33}\) specify the \(\varepsilon\)-constraints. Eq. \(\eqref{eq:34}\) represents the sum of the normal supply allocation percentages of suppliers, and Eq. \(\eqref{eq:35}\) imposes a nonnegative constraint on all decision variables.

4.1. Simulation Process

The model is developed using AnyLogic, a visual library-based simulation software with a three-phase and job-driven approach, enabling detailed and complex process modeling ³⁹. The model inputs and outputs are shown in Fig. 3, and the simulation proceeds through the following major steps: (1) predicting demand for the entire simulation period, (2) determining expected procurement time, (3) handling demand arrival and customer order placement, (4) calculating flexibility degree and safety stock ratio, (5) determining delayed orders and lost sales, (6) calculating and placing negotiated backup supply, (7) evaluating average delivery time, plan instability, and nervousness KPIs, (8) calculating and placing primary backup supply, (9) determining actual and anticipated inventory levels and normal supply required, (10) splitting and placing supply orders based on allocation percentages considering capacity constraints and uncertain procurement times, (11) calculating total profit, and (12) determining the averages of KPIs and performance measures. Steps 2–11 repeat in each planning cycle as the planning horizon rolls forward by one period (month) throughout the simulation length. Step 12 is performed at the end of each simulation run of the optimization iterations.

4.2. Optimization Process

Among the available optimization engines in AnyLogic, OptQuest is selected for this study because it combines scatter search, Tabu Search, and neural networks as primary search strategies, enabling it to efficiently handle the complex and multidimensional nature of the integrated S&OP problem and achieve faster convergence ⁴⁰.

Figure 3 also shows the interaction between the simulation and optimization engines. The optimization engine first generates decision-making variables by considering their bounds, requirements, and suggested values. These variables, together with the input parameters, are entered into the simulation model to determine the KPIs. Each simulation run is repeated multiple times to account for demand and procurement time uncertainty. The optimization engine calculates the average of the KPIs from these replications and checks them against the \(\varepsilon\)-constraint limits. If any constraint is violated, the solution is deemed infeasible. Otherwise, the optimization engine compares the total profit from the current iteration with the previous best value. If a higher profit is found, it updates the best objective value and the corresponding solution. In the next iteration, the optimization engine searches for other decision-making variables in the solution space that can potentially maximize the total profit compared with the current best value. This iterative process continues until the maximum number of iterations is reached and an optimal solution is found. Whenever an iteration improves the primary objective, the decision variables, total profit, \(\varepsilon\)-constraint values, and key performance measures are recorded.

4.3. Industrial Case Study and Experimental Design

In our previous study ⁹, the Bishoftu Automotive Industry (BAI) case was used to contextualize and apply an S&OP model from the literature using simulation. In the present study, the case is updated with qualitative and quantitative data and used to evaluate the integrated S&OP simulation–optimization framework proposed. The updates include information on delivery time delays, rescheduling practices, customer behavior in response to backorders, key suppliers, and their attributes, including capacity, procurement lead-time and uncertainty, and unit supply costs.

BAI, an Ethiopian company, purchases automotive parts mainly from Chinese suppliers and sells assembled light pickup trucks, buses, and heavy-duty trucks in the local market. The double-cabin light pickup truck, which has a diverse customer base with small order quantities, is selected for this case study. The marketing and sales department prepares an annual demand plan with monthly breakdowns based on qualitative sales analysis and market research. The logistics department conducts international bids to select suppliers and procures parts once foreign currency exchange is secured. The procurement is carried out in CKD units, primarily from two major suppliers that have established long-term relationships with the company. The procurement lead times are long and uncertain with a uniform distribution. Customer orders are registered in advance while procurement of parts is underway.

The major challenges include high demand forecast errors, long and uncertain procurement lead times, and delayed deliveries. Extended procurement lead times result from international bidding processes and delays in securing foreign currency. Facing these supply constraints, the company adopts a risk-averse strategy of registering customer orders but postponing delivery commitments until parts physically arrive and repeatedly revising production plans based on the availability of parts. This often leads to low capacity utilization, delayed delivery times, and price fluctuations, which may lead to customer order cancelations.

The historical sales data show no significant trend or seasonality and follow a uniform distribution. The average customer demand is assumed to equal the full capacity of two 8-hr shifts per day throughout the month due to low capacity utilization. The collected company data include forecasting error, procurement lead times, normal supply cost per unit, inventory holding cost per unit per period, normal time manufacturing capacity, normal time manufacturing cost per unit, overtime-to-normal time manufacturing cost per unit ratio, and reference price. The relationship between the frozen horizon length and lost sales threshold, backup supply cost per unit, and supplier capacities are estimated for the study. The other input parameters are provided in Table 4. Due to confidentiality concerns, only some of the input parameters are presented in the table.

Three experiments are designed and conducted to examine the effects of frozen horizon length adjustments and evaluate the impact of supply-side factors on system performance and plan stability objectives. The experiments further analyze the dynamic interactions among decision variables, internal (frozen horizon length, sourcing policy, and supply interval) and external (supplier capacity constraint, unit cost, and lead-time uncertainty) factors, and the trade-offs among objectives within the integrated model. The maximum values for the \(\varepsilon\)-constraints in all experiments are set as \(\textit{DOP}^{\textit{max}}=4.5\), \(\textit{LSP}^{\textit{max}}=2.5\), \(T^{\textit{max}}=4.25\), \(I^{\textit{max}}=25\), \(\textit{LN}^{\textit{max}}=0.980\), and \(\textit{GN}^{\textit{max}}=0.960\). The experiments and their corresponding descriptions are presented as follows:

1. Experiment 1: Effect of Frozen Horizon Length In this experiment, tests are conducted for frozen horizon lengths of 1–7 to evaluate their effects on system performance and plan stability. This also helps determine the appropriate length for use in subsequent experiments. The flexible horizon length is set to 5, causing the total planning horizon to increase with each frozen horizon increment. The absolute and relative flexibility degrees and safety stock ratios are the decision-making variables. Supplier 1, with a procurement lead time of \(L_{1t}=[6,10]\), is used without capacity restriction.

2. Experiment 2: Sourcing Policies In this experiment, the capacity constraints of Supplier 1 are applied, and Supplier 2 with a procurement time of \(L_{2t}=[6,8]\) is also utilized. However, the normal supply cost per unit of Supplier 2 is 1.5% higher than that of Supplier 1. The capacity of Supplier 1 is \({Ca}_1=[30,90]\), whereas that of Supplier 2 is unconstrained. Thus, four sourcing policies are designed and tested. Policy 1: Supplier 1 without capacity constraints; Policy 2: Supplier 1 under capacity constraints; Policy 3: Supplier 2 without capacity constraints; and Policy 4: bi-sourcing, where both Supplier 1 with capacity constraints and Supplier 2 without constraints are utilized. Here, the order allocation percentages (\(q_i\)) are considered as additional decision-making variables.

3. Experiment 3: Sensitivity to Normal Supply Interval and Procurement Time Uncertainty The sensitivity of the S&OP system is tested for three procurement time uncertainty levels under three NSI values. The procurement uncertainty levels are as follows. Deterministic: \(L_{1t}/L_{2t}=8/8\); Medium: \(L_{1t}/L_{2t}=[6,10]/[6, 8]\); and High: \(L_{1t}/L_{2t}=[4,12]/[4,10]\). Each uncertainty level is tested for NSI values of 1, 2, and 3. The S&OP system with the best-performing sourcing policy from Experiment 2 is used in this experiment.

5.1. Effect of Frozen Horizon Length

The results of Experiment 1 are summarized in Table 5 and Fig. 4. The table shows the freeze proportion for each frozen horizon length, defined as the percentage of frozen periods relative to the total planning horizon. The flexibility degree (\(\textit{FR}_t\)) decreased from its highest value at \(F=1\) to its lowest at \(F=6\), while the safety stock ratio (\(\textit{SS}_t\)) remained nearly zero, except at \(F=7\). As \(F\) increased from 1 to 7, the production order quantity (\(N_t\)) and normal supply arrivals (\(n_{1t}\)) decreased by 1.8% and 1.1%, respectively. Primary backup supply and actual inventory levels declined by 5.9% and 11.7%, respectively, whereas negotiated backup supply increased by 38.1%.

Figure 4 shows the objective values, with the error bars representing the standard deviations. The error bars of the average delivery time are nearly invisible due to its very low variability. Overall, variability across replications was low, with a coefficient of variation below 10% for most objectives, indicating stable simulation performance. The lost sales exhibited a relatively higher coefficient of variation due to its small mean value.

As shown in Fig. 4(a), the total profit in Ethiopian Birr (ETB) did not change significantly across all frozen horizon lengths (\(p=0.459\)), suggesting that it remains largely unaffected by freezing decisions. By contrast, delayed orders decreased substantially by 63.4% as \(F\) increased beyond 2, as shown in Fig. 4(b). Lost sales increased initially by 34.8% at \(F=2\) and remained stable through \(F=5\) before reaching its highest level at \(F=7\). This pattern suggests that excessive freezing shifts most unmet demand toward lost sales as customer impatience increases. Delivery time generally increased with longer frozen horizons, and only minimal increments were observed from \(F=2\) to 3 and \(F=3\) to 4 (Fig. 4(c)). Instability declined by 29.6% from \(F=1\) to 7. Local nervousness (Fig. 4(d)) similarly decreased by 2.6% beyond \(F=2\). In contrast, global nervousness increased by 1.0% from \(F=1\) to 3, remained stable through \(F=5\) (\(p=0.489\)), and then decreased slightly toward \(F=7\). These results indicate that delayed orders exhibited a pattern similar to local nervousness, while lost sales mirrored the trends of global nervousness.

Overall, longer frozen horizons reduced delayed orders, instability, and local nervousness without significantly affecting total profit. However, these improvements came at the expense of responsiveness, as reflected in higher lost sales, longer delivery times, and slightly elevated global nervousness. To balance stability and responsiveness, \(F=3\) was selected from the reasonable range (\(F=2\)–5) and used in the subsequent experiments.

5.2. Sourcing Policies

Table 6 presents the decision variables, key performance measures, and optimal objective values. Under Policy 2, the flexibility degree (\(\textit{FR}_t\)) decreased significantly, while the safety stock ratio (\(\textit{SS}_t\)) increased compared to Policy 1. The production order quantity (\(N_t\)) showed no significant change (\(p=0.371\)). The normal supply requirement (\(n_t\)) increased substantially, while arrivals (\(n_{1t}\)) declined due to capacity constraints. Consequently, backup supply cost increased, reducing total profit by 3.9% despite lower normal supply and inventory holding costs. Delayed orders and plan instability increased by 12.2% and by 2.1%, respectively. By contrast, lost sales, average delivery time, and local and global nervousness exhibited no significant changes (\(p=0.134\), 0.793, 0.271, and 0.338, respectively).

Under Policy 3, the flexibility degree declined relative to Policy 1, but less sharply than in Policy 2. The safety stock ratio, production order quantity, and both normal supply requirement and arrivals showed no significant changes. The normal supply cost increased, whereas the inventory holding cost decreased significantly. The backup supply cost, manufacturing cost, and total sales exhibited no significant changes (\(p=0.139\), 0.724, and 0.726, respectively). The decline in total profit (5.6%) was driven solely by the higher unit supply cost of Supplier 2, rather than by backup supply or inventory holding costs. Delayed order percentage and plan instability increased by 6.1% (\(p=0.016\)) and 0.8% (\(p=0.014\)), respectively. The lost sales percentage, average delivery time, and both local and global nervousness remained unchanged (\(p=0.771\), 0.338, 0.113, and 0.091, respectively).

The bi-sourcing policy (Policy 4) showed the lowest flexibility degree and minimal safety stock ratio among all policies. Normal supply was allocated 68% to Supplier 1 and 32% to Supplier 2, with arrivals of 44.3 (33.7% of \(n_t\)) from Supplier 1 and 42.1 (32.0% of \(n_t\)) from Supplier 2, indicating that \(q_2\) was fully realized, whereas \(q_1\) was only partially fulfilled. The production quantity remained unchanged across all policies.

Compared to Policies 2 and 3, the bi-sourcing policy increased total profit by 4.8% and 6.7%, respectively. Delayed orders were unchanged relative to Policy 2 (\(p=0.302\)) but increased by 8.0% compared to Policy 3. Lost sales decreased significantly by 8.8% (\(p=0.006\)) and 5.3% (\(p=0.050\), actual value \(=0.0498\)), while delivery time shortened slightly by 0.05% compared to both policies (\(p=0.014\) and 0.002). Plan instability increased by 1.0% (\(p=0.009\)) and 2.3%, whereas both local and global nervousness declined by 0.3% and 0.2% relative to both policies. Compared to Policy 1, Policy 4 also improved total profit by 0.7% (\(p=0.015\)), shortened delivery time by 0.05% (\(p=0.022\)), and reduced local (\(-\)0.4%) and global (\(-\)0.2%) nervousness. Lost sales remained unchanged (\(p=0.163\)), while delayed orders and instability increased by 14.5% and 3.1%, respectively.

5.3. Sensitivity to Normal Supply Interval and Procurement Time Uncertainty

Table 7 presents the sensitivity analysis of procurement time uncertainty levels under different normal supply intervals (NSIs) using the bi-sourcing policy from Experiment 2. For improved readability, the cost and profit data in the table are reported in thousands. At \(\textit{NSI}=1\) and 3, the flexibility degree increased with greater uncertainty, while the safety stock ratio declined, indicating a shift to more responsive planning. At \(\textit{NSI}=2\), however, the highest flexibility degree and lowest safety stock ratio occurred at the medium uncertainty level, suggesting a nonlinear response to uncertainty. The production order quantity showed limited sensitivity overall, declining only at specific NSI–uncertainty combinations, such as at \(\textit{NSI}=1\) under medium (\(p=0.004\)) and high uncertainty, and at \(\textit{NSI}=3\) under medium uncertainty (\(p=0.010\)). Supplier allocation shifted with longer supply intervals, with \(q_1\) near 100% at \(\textit{NSI}=1\) and increased \(q_2\) shares at \(\textit{NSI}=2\) and 3. Correspondingly, the arrival coverage (\(n_{1t}\)) of Supplier 1 declined as NSI increased, while Supplier 2 provided the highest coverage under the deterministic case at \(\textit{NSI}=3\).

At \(\textit{NSI}=1\), total profit declined by 3.7% under medium uncertainty due to higher backup supply costs and excess inventory holding costs, and it decreased further by 1.2% under high uncertainty. At \(\textit{NSI}=2\), it remained statistically unchanged (\(p=0.128\)) under medium uncertainty despite higher backup supply and inventory holding costs but declined by 1.2% under high uncertainty. By contrast, at \(\textit{NSI}=3\), profit increased by 1.9% under medium uncertainty because of reduced holding costs before declining by 1.5% under high uncertainty as reliance on backup supply increased due to shortages from the longer supply interval and sparse arrivals.

Delayed orders remained largely stable across uncertainty levels, with reductions observed only at \(\textit{NSI}=1\) (\(-\)5.8%, high vs. deterministic) and at \(\textit{NSI}=3\) (\(-\)8.6%, high vs. medium; \(-\)8.4%, high vs. deterministic). Lost sales increased significantly at \(\textit{NSI}=1\) by 25.1% under medium and 9.3% (\(p=0.007\)) under high uncertainty levels, and at \(\textit{NSI}=3\) by 12.3% and 11.1% (\(p=0.002\)) under medium and high uncertainty compared to the deterministic case, respectively. At \(\textit{NSI}=2\), only the medium level exceeded the deterministic case (7.3%, \(p=0.014\)). Average delivery time increased by 0.05% at both \(\textit{NSI}=1\) and 3 (\(p=0.025\) and \(p=0.004\)) under high uncertainty compared to the deterministic case. Plan instability generally declined as uncertainty rose, with reductions of 2.3% (medium vs. deterministic) and 1.0% (high vs. medium; \(p=0.006\)) at \(\textit{NSI}=1\), and 1.6% (high vs. medium) at \(\textit{NSI}=3\). By contrast, local and global nervousness increased significantly with uncertainty, except at \(\textit{NSI}=2\), where the differences were not significant.